Miniproject_602

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Miniproject-MRC602 2010

DuyKhanh Pham Page 1

Mobile Radio Communications 602 (11333 v.4) (Semester 2, 2010)

Objective

In this laboratory, we use Matlab to simulate a multipath radio channel characterised by

Rayleigh fading and estimate effects of velocity on a fading channel.

Background

For most channels, where signal propagate in the atmosphere and near the ground, the

free-space propagation model is inadequate to describe the channels behaviour and predict

system performance. In wireless system, s signal can travel from transmitter to receiver over

multiple reflective paths. This phenomenon, called multipath fading, can cause fluctuations in

the received signals amplitude, phase, and angle of arrival, giving rise to the terminology

multipath fading. Another name, scintillation, is used to describe the fading caused byphysical changes in the propagating medium, such as variations in the electron density of the

ionosphere layers that reflect high frequency radio signals. Both fading and scintillation refer

to a signals random fluctuations.

The termsslow andfastfading refer to the rate at which the magnitude and phase changeimposed by the channel on the signal changes. The coherence time is a measure of theminimum time required for the magnitude change of the channel to become uncorrelated

from its previous value.

ySlow fading arises when the coherence time of the channel is large relative to thedelay constraint of the channel. In this regime, the amplitude and phase change imposed

by the channel can be considered roughly constant over the period of use. Slow fading can


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be caused by events such as shadowing, where a large obstruction such as a hill or large

building obscures the main signal path between the transmitter and the receiver. The

amplitude change caused by shadowing is often modeled using a log-normal distribution

with a standard deviation according to the log-distance path loss model.

yFast fading occurs when the coherence time of the channel is small relative to thedelay constraint of the channel. In this regime, the amplitude and phase change imposed

by the channel varies considerably over the period of use.

In a fast-fading channel, the transmitter may take advantage of the variations in thechannel conditions using time diversity to help increase robustness of the communication to a

temporary deep fade. Although a deep fade may temporarily erase some of the informationtransmitted, use of an error-correcting code coupled with successfully transmitted bits during

other time instances (interleaving) can allow for the erased bits to be recovered. In a slow-

fading channel, it is not possible to use time diversity because the transmitter sees only a

single realization of the channel within its delay constraint. A deep fade therefore lasts the

entire duration of transmission and cannot be mitigated using coding.

The coherence time of the channel is related to a quantity known as the Doppler spread of

the channel. When a user (or reflectors in its environment) is moving, the user's velocity

causes a shift in the frequency of the signal transmitted along each signal path. This

phenomenon is known as the Doppler shift. Signals travelling along different paths can have

different Doppler shifts, corresponding to different rates of change in phase. The difference in

Doppler shifts between different signal components contributing to a single fading channel

tap is known as the Doppler spread. Channels with a large Doppler spread have signal

components that are each changing independently in phase over time. Since fading dependson whether signal components add constructively or destructively, such channels have a very

short coherence time.

As the carrier frequency of a signal is varied, the magnitude of the change in amplitudewill vary. The coherence bandwidth measures the separation in frequency after which two

signals will experience uncorrelated fading.

yIn flat fading, the coherence bandwidth of the channel is larger than the bandwidth ofthe signal. Therefore, all frequency components of the signal will experience the same

magnitude of fading.

yIn frequency-selective fading, the coherence bandwidth of the channel is smaller thanthe bandwidth of the signal.

Since different frequency components of the signal are affected independently, it is highly

unlikely that all parts of the signal will be simultaneously affected by a deep fade. Certain

modulation schemes such as OFDM and CDMA are well-suited to employing frequency

diversity to provide robustness to fading. OFDM divides the wideband signal into manyslowly modulated narrowband subcarriers, each exposed to flat fading rather than frequency

selective fading. This can be combated by means of error coding, simple equalization oradaptive bit loading. Inter-symbol interference is avoided by introducing a guard interval

between the symbols. CDMA uses the Rake receiver to deal with each echo separately.


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Result & Calculation

Task 1

Plot a graph of the CDF against the signal level in dB relative to the RMS value. Show

that the fading envelope fluctuates over a range from 30 dB to 10 dB with a probability of

99.9 %.

Figure1

From the graph, the CDF fluctuates from -30dB to 10dB increasing gradually with

probability 99.99%.

Task2

A mobile station operating with a carrier frequency of 900 MHz is traveling at 54 km/hr.

Plot a graph of the ratio of LCR to maximum Doppler shift against the signal level in dB

relative to its rms value. Show that the maximum LCR occurs at = -3dB with a value of

about 48 per second.


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Figure2

According to the curve, we see that the ratio has the value 48 per second at p = -3dB. This

mean that the rate at the particular level R crossed in positive direction is 48 per second at

threshold -3dB.

Task3

Plot a graph of the product of AFD and maximum Doppler shift against the signal level in

dB relative to the rms value. Show that the ADF is around 8 ms for = -3dB.

Figure3


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We observe that AFD is zero second up to the initial signal level (around -13dB) but

increase rapidly when the signal level is greater than 5 dB. From Figure3, (AFD*fd) is

around 0.3667 seconds at signal level p = -3dB. In this case, we use v = 54km/hr, c = 3*10^8

m/s, fc = 900 Mhz => fd = v * fc / c = 45 m/s. Therefore, we get AFD is 0.3667 / 45 =

0.00815 seconds at p = -3dB that agrees to theoretical value (8ms).

Task 4

Figure4

This figure shows the Filter coefficients varying in signal level over N points. When k = 0,

the output Fk is zero then increasing gradually from k = 1, and reach a peak 3dB at k = km

then falls down to zero from (km+1) to (N-km-1). The output get a peak again at (N-km) then

goes down to zero. This is because of the following equation we use to generate Fk:


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Next we plot the amplitude fading sequence Y[n]

Figure5

From the result, we can see the amplitude of Y signal fluctuates from 0 to 3dB. The

average amplitude is around 1dB.


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Figure6

Figure6 show the phase distortion of Y sequence. We observe that the phase distortion

varies from -3dB to 3dB. This range is almost constant during N points. Now we plot the

empirical Probablity density function (PDF) and its theoretical one (blue line) in the same

graph.

Figure7


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The PDF increases gradually from 0dB to the maximum 0.85dB then drops down to zero.

Compare to theory, the empirical PDF is a bit fluctuation which depends on the min and

max of Y amplitude. However, the average value is close to theoretical one.

Figure8

Figure8 shows empirical PDF versus theoretical PDF of phase distortion Y signal. We cansee that the theoretical PDF of Y is straight line with constant value 0.16dB from range of -

3dB to 3dB whereas the empirical PDF varies from 0.02dB to 0.37dB. However, it could be

0.15db in average which is close to theory.

This section we find the empirical LCR and AFD for the amplitude fading sequence Y.

Figure9 below shows the empirical and theoretical is not match together from -30dB to

0dB. However, they get the same peak 45dB at -3dB.


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Figure9

Next we plot the theoretical and practical AFD of Y amplitude signal

Figure10

From the figure above, we see that two lines are match from -40dB to 5dB. After that, the

practical AFD increases rapidly to reach a peak 65dB at 10 dB on X axis while the empirical

AFD stays at the level 5 dB.


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For v =5km/hr

Figure11

Figure11 shows the Filter coefficients. The difference between two cases is that the gap

between two peaks is bigger and the maximum value is 1.7dB compare to the peak in

previous case just 3dB.

Figure12

We see that the amplitude variation between each period is not as dense as the one in case

velocity 50km/hr.


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Figure13

Although the phase distortion is distinctive compare to last case but it steel has the same

value, varies from -3dB to 3 dB and almost constant over rang of N points.

Figure14

From figure14, the empirical PDF is not close to theoretical PDF like in case v = 50km/hr.

It is more fluctuated.


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Figure15

For the phase distortion, we observe that the empirical PDF varies randomly.

Figure16

There are big gaps between empirical PDF and practical PDF for LCR. We can say the

matching between them in the first case as much better than this case. Moreover, from the

graph, the peak of LCR in this case is just 4.5 dB compare to 45dB in case v = 50km/hr.


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Figure17

From figure17, the empirical AFD is almost zero all time while the practical AFD is stable

at zero dB from -40dB to 5dB then increase rapidly up to 650dB (65 when v = 50km/hr).

After changing the velocity from 50km/hr to 5km/hr, the Doppler frequency is reduced 10

times as fd = (v * fc / c).

LRC equation:

AFD equation:

According to these equations, when fd reduces 10 times, it causes LRC decreases 10 times

and AFD increases 10 times as well.


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Discussion

Task1: Because Rayleigh fading is an example of flat fading, the received signal varies in

amplitude but the spectrum of the transmitted signal is preserved, therefore we get the fading

envelope fluctuates from a range from -30dB to 10dB with a probability of 99.99%.

Task2: As we know, LCR is defined by the expected rate crossed a particular level R in a

upward direction. Fading signal crossing in negative direction is not counted as they do not

carry any information. For example, on task 2 we get LCR is 48/sec at p = -3dB. That means

the level R is crossed at rate of 48 per second.

Task3: This task we learn about average fading duration AFD. We look for the fad

duration corresponding to -3dB level. In this case, we get the duration for which the fading

level is below level R is 8 msec.

Task4: We write a computer program in Matlab, to simulate the complex channel gain of theRayleigh fading channel as shown in Figure below:

We implement the IDFT block using the inverse fast Fourier transform (IFFT) algorithm.

Firstly, we generate the filter coefficient. Then signals A and B are obtained by Gaussian

sequences. Y signal is verified by checking its mean = zero and variance = 0.5. According to

definitions of LCR and AFD, LCR is set when (Y_amp(j))R(i)while

AFD is set when (Y_amp(j))


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Matlab code

% Task 1:

clc

clearall;RdB = -30 : 0.1 : 10;

R = 10.^(RdB/20);

Pr = 1 - exp(-R.^2);plot(RdB,Pr,'r');

xlabel('Signal level in dB');

ylabel('CDF');

% Task 2clc

clearall;RdB = -30 : 0.1 : 10;

R = 10.^(RdB/20);c = 3*10^8;

fc = 900*10^6;v = 54*10^3/3600;

fd = v*fc/c;

NR = sqrt(2*pi)*fd*R.*exp(-R.^2);

plot(RdB,NR,'r');

xlabel('signal level in dB');

ylabel('Level crossing rate');

% Task 3

clc

clearall;

RdB = -30 : 0.1 : 10;

R = 10.^(RdB/20);c = 3*10^8;

fc = 900*10^6;v = 54*10^3/3600;

fd = v*fc/c;

Tr = (exp(R.^2)-1)./(sqrt(2*pi)*fd*R);plot (RdB, Tr*fd,'r');

xlabel('signal level in dB');ylabel('The product of AFD and maximum Doppler shift');

% Task4

% For v = 50km/hr;

clearall;

clc;

b = 4797;


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u = 6925;

v = 50*10^3/3600;

fc = 900*10^6;

c = 3*10^8;

Ts = 10^-3;

N = 2^12;n = 2^18;

fd = v*fc/c;

fm = fd*Ts;

km = floor(fm*N);

X1(1) = 4;

X2(1) = 8;

form = 1 : (N-1);X1(m+1) = mod((b*X1(m) + u),(n));

X2(m+1) = mod((b*X2(m) + u),(n));end

% Generate real strings for A & B (/n to make the value inside log less than 1)

A = sqrt(-2*log(X1/n)).*cos(2*pi*X2/n);B = sqrt(-2*log(X1/n)).*sin(2*pi*X2/n);

% Fk

F(1) = 0;

fork = 2 : km;

F(k) = sqrt(1/(2*sqrt(1-(k/(km+1))^2)));

end

F(km+1) = sqrt((km+1)/2*(pi/2 -atan(km/sqrt(2*km+1))));

fork = (km+2) : N - km;

F(k) = 0;

end

F(N - km +1) = sqrt((km+1)/2*(pi/2 -atan(km/sqrt(2*km+1))));fork = (N - km +2) : N;

F(k) = sqrt(1/(2*sqrt(1-((N-k)/(km+1))^2)));

endfigure(1);

plot(F,'r');Xlabel('N points');

Ylabel('dB');title('Plotting F');

ZI = A.*F;

ZQ = B.*F;Zk = ZI - j.*ZQ;

Z = ifft(Zk);

Z_real = real(Z);

Z_img = imag(Z);

sigma_y = sqrt(sum(F.^2)/(N^2));

Y = Z./(sigma_y*sqrt(2));

Y_real = real(Y);


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Y_img = imag(Y);

Y_amp = sqrt((Y_real.^2)+(Y_img.^2));

Y_phase = atan2(Y_img,Y_real);

figure(2);

plot(Y_amp,'g');Xlabel('N points');

Ylabel('The amplitude fading sequence Y in dB');

title('Plotting amplitude of Y');

figure(3);

plot(Y_phase,'b');

Xlabel('N points');Ylabel('The phase distortion of sequence Y in dB');

title('Plotting phase of Y');

bins = 500;

Y_amp = reshape(Y_amp, numel(Y_amp),1);ifisreal (Y_amp),

Y_amp = abs(Y_amp);

end

minY_amp = min(Y_amp);

maxY_amp = max(Y_amp);

ifminY_amp == maxY_amp

plot(minY_amp,1);

else

step = (maxY_amp - minY_amp)/(bins-1);

binc = minY_amp : step : maxY_amp;hist_amp = hist(Y_amp,bins);

pdf_amp = (hist_amp*bins)/(N*(maxY_amp-minY_amp));

endfigure(4);

plot(binc,pdf_amp,'r');xlabel('Amplitude range of Y_amp');

ylabel('dB');hold on

% Theory

sigma_theo = 0.5;

pdf_amp_theo = (binc/sigma_theo).*exp((-binc.^2)/(2*sigma_theo));plot(binc,pdf_amp_theo);

xlabel('Amplitude range of Y_amp');

ylabel('dB');

hold off

% Plotting empirical pdf phase

step = (max(Y_phase) - min(Y_phase))/(bins-1);

binc = min(Y_phase) : step : max(Y_phase);


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hist_phase = hist((Y_phase),bins);

pdf_phase = (hist_phase*bins)/(N*(max(Y_phase)-min(Y_phase)));

figure(5);

plot(binc,pdf_phase,'r');

xlabel('Phase range of Y_phase');

ylabel('dB');hold on

% Theory

pdf_phase_theo = 1/(2*pi);

plot(binc, pdf_phase_theo);

hold off

% % % AFD & LCRxdB = -30:1:10;

r = 10.^(xdB./20);

Rrms = sqrt(mean(Y_amp).^2);R = min(Y_amp):0.01:max(Y_amp); % R = Rrms.*r;avg_AFD1 = zeros(1,length(R));

LCR_count=zeros(1,length(R));

uii = zeros(1,length(Y_amp));

fori = 1:length(R)

forj= 1:1:(length(Y_amp)-1)

if((Y_amp(j))R(i)

LCR_count(i) = LCR_count(i)+1;

end

end

end

fori = 1:length(R)forj= 1:1:(length(Y_amp)-1)

if(Y_amp(j))


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% % % Plotting AFD theory

AFD = (exp(r.^2)-1)./(sqrt(2*pi)*fd.*r);

figure(7);

plot (xdB, AFD, 'b');

xlabel('time period')

ylabel('AFD')hold on

% Practical AFD

plot(mag2db(R),(avg_AFD),'r')

hold off

% For v = 5km/hr

clearall;clc;

b = 4797;

u = 6925;

v = 5*10^3/3600;

fc = 900*10^6;

c = 3*10^8;

Ts = 10^-3;

N = 2^12;

n = 2^18;

fd = v*fc/c;

fm = fd*Ts;

km = floor(fm*N);X1(1) = 4;

X2(1) = 8;

form = 1 : (N-1);X1(m+1) = mod((b*X1(m) + u),(n));

X2(m+1) = mod((b*X2(m) + u),(n));end

% Generate real strings for A & B (/n to make the value inside log less than 1) A = sqrt(-2*log(X1/n)).*cos(2*pi*X2/n);

B = sqrt(-2*log(X1/n)).*sin(2*pi*X2/n);

% Fk

F(1) = 0;fork = 2 : km;

F(k) = sqrt(1/(2*sqrt(1-(k/(km+1))^2)));

end

F(km+1) = sqrt((km+1)/2*(pi/2 -atan(km/sqrt(2*km+1))));

fork = (km+2) : N - km;

F(k) = 0;

end

F(N - km +1) = sqrt((km+1)/2*(pi/2 -atan(km/sqrt(2*km+1))));


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fork = (N - km +2) : N;

F(k) = sqrt(1/(2*sqrt(1-((N-k)/(km+1))^2)));

end

figure(1);

plot(F,'r');

Xlabel('N points');Ylabel('dB');

title('Plotting F');

ZI = A.*F;

ZQ = B.*F;

Zk = ZI - j.*ZQ;Z = ifft(Zk);

Z_real = real(Z);Z_img = imag(Z);

sigma_y = sqrt(sum(F.^2)/(N^2));Y = Z./(sigma_y*sqrt(2));Y_real = real(Y);

Y_img = imag(Y);

Y_amp = sqrt((Y_real.^2)+(Y_img.^2));

Y_phase = atan2(Y_img,Y_real);

figure(2);

plot(Y_amp,'g');

Xlabel('N points');

Ylabel('The amplitude fading sequence Y in dB');

title('Plotting amplitude of Y');

figure(3);

plot(Y_phase,'b');

Xlabel('N points');Ylabel('The phase distortion of sequence Y in dB');

title('Plotting phase of Y');

bins = 500;

Y_amp = reshape(Y_amp, numel(Y_amp),1);ifisreal (Y_amp),

Y_amp = abs(Y_amp);

endminY_amp = min(Y_amp);

maxY_amp = max(Y_amp);

ifminY_amp == maxY_amp

plot(minY_amp,1);

else

step = (maxY_amp - minY_amp)/(bins-1);

binc = minY_amp : step : maxY_amp;


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hist_amp = hist(Y_amp,bins);

pdf_amp = (hist_amp*bins)/(N*(maxY_amp-minY_amp));

end

figure(4);

plot(binc,pdf_amp,'r');

xlabel('Amplitude range of Y_amp');ylabel('dB');

hold on

% Theory

sigma_theo = 0.5;

pdf_amp_theo = (binc/sigma_theo).*exp((-binc.^2)/(2*sigma_theo));

plot(binc,pdf_amp_theo);xlabel('Amplitude range of Y_amp');

ylabel('dB');hold off

% Plotting empirical pdf phasestep = (max(Y_phase) - min(Y_phase))/(bins-1);

binc = min(Y_phase) : step : max(Y_phase);

hist_phase = hist((Y_phase),bins);

pdf_phase = (hist_phase*bins)/(N*(max(Y_phase)-min(Y_phase)));

figure(5);

plot(binc,pdf_phase,'r');

xlabel('Phase range of Y_phase');

ylabel('dB');

hold on

% Theorypdf_phase_theo = 1/(2*pi);

plot(binc, pdf_phase_theo);

hold off

% % % AFD & LCRxdB = -30:1:10;

r = 10.^(xdB./20);Rrms = sqrt(mean(Y_amp).^2);

R = min(Y_amp):0.01:max(Y_amp); % R = Rrms.*r;avg_AFD1 = zeros(1,length(R));

LCR_count=zeros(1,length(R));

uii = zeros(1,length(Y_amp));fori = 1:length(R)


if((Y_amp(j))R(i)

LCR_count(i) = LCR_count(i)+1;

end

end

end

fori = 1:length(R)


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if(Y_amp(j))

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